We start a systematic study of general group valuations. A valuation is a definite function from a (generally non-Abelian) group to an ordered set with a least element mapping the neutral element to the least element and satisfying the ultrametric inequality.
In Section 2 the definition of a general group valuation is given and the basic properties of these valuations are studied. The main result of this section is a theorem which gives several different criteria for a valuation to make the group a topological group. Furthermore, we introduce various types of valuations, such as topological, uniformly topological, normal, conjugation isotonic or conjugation invariant, and study the relations between them. Quotient groups are examined in Section 3. Here the non-surjectivity of the valuation allows for certain natural results which may not been obtained otherwise. Sequences and subsequences are introduced in Section 4. Here we also present some probably new results on cofinality types of ordered sets. We use them to characterize subsequences and index sets of Cauchy sequences. The theory of pseudo- convergence is set forth in Section 5. Particularly it is shown that a right proper Cauchy sequence has a right pseudo-Cauchy subsequence. In the course of the proof the earlier results about the cofinality types are needed.